Let x0 of type (ι → ι → ι → ι) → (ι → ι → (ι → ι) → ι) → ι be given.
Let x1 of type ((CT3 ι) → (CT2 ι) → ι) → ι → ι be given.
Let x2 of type (ι → ι) → ι → ι → ι be given.
Let x3 of type (ι → ι → (ι → ι → ι) → (ι → ι) → ι → ι) → (ι → ι) → ι → ι be given.
Assume H0:
∀ x4 x5 x6 . ∀ x7 : ι → ι → ι → ι → ι . x3 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . setsum 0 (x3 (λ x13 x14 . λ x15 : ι → ι → ι . λ x16 : ι → ι . λ x17 . x17) (λ x13 . setsum (x1 (λ x14 : (ι → ι → ι → ι) → ι . λ x15 : (ι → ι → ι) → ι . 0) 0) x13) (x2 (λ x13 . x13) (x2 (λ x13 . 0) 0 0) (x2 (λ x13 . 0) 0 0)))) (λ x8 . x5) (x3 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . x0 (λ x13 x14 x15 . Inj1 (x2 (λ x16 . 0) 0 0)) (λ x13 x14 . λ x15 : ι → ι . 0)) (λ x8 . setsum (x1 (λ x9 : (ι → ι → ι → ι) → ι . λ x10 : (ι → ι → ι) → ι . x0 (λ x11 x12 x13 . 0) (λ x11 x12 . λ x13 : ι → ι . 0)) x5) (x1 (λ x9 : (ι → ι → ι → ι) → ι . λ x10 : (ι → ι → ι) → ι . 0) (Inj0 0))) x5) = x3 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . Inj1 x12) (λ x8 . setsum 0 (Inj0 x8)) (x3 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . 0) (λ x8 . 0) x6).
Assume H1:
∀ x4 : ι → ((ι → ι) → ι) → ι → ι → ι . ∀ x5 x6 x7 . x3 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . x10 (Inj0 (setsum (x0 (λ x13 x14 x15 . 0) (λ x13 x14 . λ x15 : ι → ι . 0)) 0)) (setsum 0 (x11 (x2 (λ x13 . 0) 0 0)))) (λ x8 . x8) (x4 (Inj1 (x2 (λ x8 . 0) 0 0)) (λ x8 : ι → ι . x6) (Inj1 x7) (x1 (λ x8 : (ι → ι → ι → ι) → ι . λ x9 : (ι → ι → ι) → ι . x7) (x4 x7 (λ x8 : ι → ι . x0 (λ x9 x10 x11 . 0) (λ x9 x10 . λ x11 : ι → ι . 0)) (Inj1 0) (setsum 0 0)))) = x4 (x1 (λ x8 : (ι → ι → ι → ι) → ι . λ x9 : (ι → ι → ι) → ι . Inj0 (x1 (λ x10 : (ι → ι → ι → ι) → ι . λ x11 : (ι → ι → ι) → ι . x10 (λ x12 x13 x14 . 0)) (Inj0 0))) 0) (λ x8 : ι → ι . Inj1 (x0 (λ x9 x10 x11 . x2 (λ x12 . 0) (setsum 0 0) 0) (λ x9 x10 . λ x11 : ι → ι . x10))) x6 (setsum (x4 (x0 (λ x8 x9 x10 . 0) (λ x8 x9 . λ x10 : ι → ι . 0)) (λ x8 : ι → ι . Inj0 x5) (x1 (λ x8 : (ι → ι → ι → ι) → ι . λ x9 : (ι → ι → ι) → ι . 0) (x3 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . 0) (λ x8 . 0) 0)) (x3 (λ x8 x9 . λ x10 : ι → ι → ι . λ x11 : ι → ι . λ x12 . setsum 0 0) (λ x8 . x5) 0)) 0).
Assume H2:
∀ x4 : (ι → (ι → ι) → ι) → ι . ∀ x5 : ι → ι . ∀ x6 x7 . x2 (λ x8 . x8) (Inj1 0) (x1 (λ x8 : (ι → ι → ι → ι) → ι . λ x9 : (ι → ι → ι) → ι . x7) (x2 (λ x8 . x6) (x1 (λ x8 : (ι → ι → ι → ι) → ι . λ x9 : (ι → ι → ι) → ι . 0) (setsum 0 0)) (x0 (λ x8 x9 x10 . x10) (λ x8 x9 . λ x10 : ι → ι . x7)))) = Inj0 (x4 (λ x8 . λ x9 : ι → ι . 0)).